A discrete weighted Markov--Bernstein inequality for polynomials and sequences

Abstract

For parameters c ∈ (0, 1) and β > 0, let ℓ 2 (c, β) be the Hilbert space of real functions defined on N (i.e., real sequences), for whichWe study the best (i.e., the smallest possible) constant γn(c, β) in the discrete Markov-Bernstein inequalitywhere Pn is the set of real algebraic polynomials of degree at most n and ∆f (x) := f (x + 1) − f (x) .We prove that(ii) For every fixed c ∈ (0, 1) , γn(c, β) is a monotonically decreasing function of β in (0, ∞) ; (iii) For every fixed c ∈ (0, 1) and β > 0 , the best Markov-Bernstein constants γn(c, β) are bounded uniformly with respect to n. A similar Markov-Bernstein unequality is proved for sequences in ℓ 2 (c, β) . We also establish a relation between the best Markov-Bernstein constants γn(c, β) and the smallest eigenvalues of certain explicitly given Jacobi matrices.